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The harmonic number Hn,1{displaystyle H_{n,1}} with n=⌊x⌋{displaystyle n=lfloor xrfloor } (red line) with its asymptotic limit γ+ln⁡(x){displaystyle gamma +ln(x)} (blue line) where γ{displaystyle gamma } is the Euler–Mascheroni constant.

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

Hn=1+12+13+⋯+1n=∑k=1n1k.{displaystyle H_{n}=1+{frac {1}{2}}+{frac {1}{3}}+cdots +{frac {1}{n}}=sum _{k=1}^{n}{frac {1}{k}}.}

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.

Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.

The harmonic numbers roughly approximate the natural logarithm function^[1]:143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.

Bertrand's postulate entails that, except for the case n = 1, the harmonic numbers are never integers.^[2]

Identities involving harmonic numbers

By definition, the harmonic numbers satisfy the recurrence relation

Hn+1=Hn+1n+1.{displaystyle H_{n+1}=H_{n}+{frac {1}{n+1}}.}

The harmonic numbers are connected to the Stirling numbers of the first kind by the relation

Hn=1n![n+12].{displaystyle H_{n}={frac {1}{n!}}left[{n+1 atop 2}right].}

The functions

fn(x)=xnn!(log⁡x−Hn){displaystyle f_{n}(x)={frac {x^{n}}{n!}}(log x-H_{n})}

satisfy the property

fn′(x)=fn−1(x).{displaystyle f_{n}'(x)=f_{n-1}(x).}

In particular

f1(x)=x(log⁡x−1){displaystyle f_{1}(x)=x(log x-1)}

is an integral of the logarithmic function.^{[citation needed]}

The harmonic numbers satisfy the series identity

∑k=1nHk=(n+1)(Hn+1−1).{displaystyle sum _{k=1}^{n}H_{k}=(n+1)(H_{n+1}-1).}

Identities involving π

There are several infinite summations involving harmonic numbers and powers of π:^[3]

∑n=1∞Hnn⋅2n=112π2{displaystyle sum _{n=1}^{infty }{frac {H_{n}}{ncdot 2^{n}}}={frac {1}{12}}pi ^{2}}

∑n=1∞Hn2(n+1)2=11360π4{displaystyle sum _{n=1}^{infty }{frac {H_{n}^{2}}{(n+1)^{2}}}={frac {11}{360}}pi ^{4}}

∑n=1∞Hn2n2=17360π4{displaystyle sum _{n=1}^{infty }{frac {H_{n}^{2}}{n^{2}}}={frac {17}{360}}pi ^{4}}

∑n=1∞Hnn3=172π4{displaystyle sum _{n=1}^{infty }{frac {H_{n}}{n^{3}}}={frac {1}{72}}pi ^{4}}

Calculation

An integral representation given by Euler^[4] is

Hn=∫011−xn1−xdx.{displaystyle H_{n}=int _{0}^{1}{frac {1-x^{n}}{1-x}},dx.}

The equality above is straightforward by the simple algebraic identity

1−xn1−x=1+x+⋯+xn−1.{displaystyle {frac {1-x^{n}}{1-x}}=1+x+cdots +x^{n-1}.}

Using the substitution x = 1 − u, another expression for H_n is

Hn=∫011−xn1−xdx=∫011−(1−u)nudu=∫01[−∑k=1n(−1)k(nk)uk−1]du=−∑k=1n(−1)k(nk)∫01uk−1du=−∑k=1n(−1)k1k(nk).{displaystyle {begin{aligned}H_{n}&=int _{0}^{1}{frac {1-x^{n}}{1-x}},dx\[6pt]&=int _{0}^{1}{frac {1-(1-u)^{n}}{u}},du\[6pt]&=int _{0}^{1}left[-sum _{k=1}^{n}(-1)^{k}{binom {n}{k}}u^{k-1}right],du\[6pt]&=-sum _{k=1}^{n}(-1)^{k}{binom {n}{k}}int _{0}^{1}u^{k-1},du\[6pt]&=-sum _{k=1}^{n}(-1)^{k}{frac {1}{k}}{binom {n}{k}}.end{aligned}}}

A closed form expression for H_n is

Hn=Gn−(n+1)⌊Gnn+1⌋{displaystyle {begin{aligned}H_{n}&=G_{n}-(n+1)leftlfloor {frac {G_{n}}{n+1}}rightrfloor end{aligned}}}

where

Gn=(n+(n+1)!n)−1(n+1)!{displaystyle {begin{aligned}G_{n}&={frac {{n+(n+1)! choose n}-1}{(n+1)!}}end{aligned}}}

Graph demonstrating a connection between harmonic numbers and the natural logarithm. The harmonic number H_n can be interpreted as a Riemann sum of the integral: ∫1n+1dxx=ln⁡(n+1).{displaystyle int _{1}^{n+1}{frac {dx}{x}}=ln(n+1).}

The nth harmonic number is about as large as the natural logarithm of n. The reason is that the sum is approximated by the integral

∫1n1xdx{displaystyle int _{1}^{n}{frac {1}{x}},dx}

whose value is ln(n).

The values of the sequence H_n − ln(n) decrease monotonically towards the limit

limn→∞(Hn−ln⁡n)=γ,{displaystyle lim _{nto infty }left(H_{n}-ln nright)=gamma ,}

where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. The corresponding asymptotic expansion as n → +∞ is

Hn∼ln⁡n+γ+12n−∑k=1∞B2k2kn2k=ln⁡n+γ+12n−112n2+1120n4−⋯,{displaystyle H_{n}sim ln {n}+gamma +{frac {1}{2n}}-sum _{k=1}^{infty }{frac {B_{2k}}{2kn^{2k}}}=ln {n}+gamma +{frac {1}{2n}}-{frac {1}{12n^{2}}}+{frac {1}{120n^{4}}}-cdots ,}

where Bk{displaystyle B_{k}} are the Bernoulli numbers.

Generating functions

A generating function for the harmonic numbers is

∑n=1∞znHn=−ln⁡(1−z)1−z,{displaystyle sum _{n=1}^{infty }z^{n}H_{n}={frac {-ln(1-z)}{1-z}},}

where ln(z) is the natural logarithm. An exponential generating function is

∑n=1∞znn!Hn=−ez∑k=1∞1k(−z)kk!=ezEin⁡(z){displaystyle sum _{n=1}^{infty }{frac {z^{n}}{n!}}H_{n}=-e^{z}sum _{k=1}^{infty }{frac {1}{k}}{frac {(-z)^{k}}{k!}}=e^{z}operatorname {Ein} (z)}

where Ein(z) is the entire exponential integral. Note that

Ein⁡(z)=E1(z)+γ+ln⁡z=Γ(0,z)+γ+ln⁡z{displaystyle operatorname {Ein} (z)=mathrm {E} _{1}(z)+gamma +ln z=Gamma (0,z)+gamma +ln z}

where Γ(0, z) is the incomplete gamma function.

Arithmetic properties

The harmonic numbers have several interesting arithmetic properties. It is well-known that Hn{textstyle H_{n}} is an integer if and only if n=1{textstyle n=1}, a result often attributed to Taeisinger.^[5] Indeed, using 2-adic valuation, it is not difficult to prove that for n≥2{textstyle ngeq 2} the numerator of Hn{textstyle H_{n}} is an odd number while the denominator of Hn{textstyle H_{n}} is an even number. More precisely,

Hn=12⌊log2⁡(n)⌋anbn{displaystyle H_{n}={frac {1}{2^{lfloor log _{2}(n)rfloor }}}{frac {a_{n}}{b_{n}}}}

with some odd integers an{textstyle a_{n}} and bn{textstyle b_{n}}.

As a consequence of Wolstenholme's theorem, for any prime number p≥5{displaystyle pgeq 5} the numerator of Hp−1{displaystyle H_{p-1}}is divisible by p2{textstyle p^{2}}. Furthermore, Eisenstein^[6] proved that for all odd prime number p{textstyle p} it holds

H(p−1)/2≡−2qp(2)(modp){displaystyle H_{(p-1)/2}equiv -2q_{p}(2){pmod {p}}}

where qp(2)=(2p−1−1)/p{textstyle q_{p}(2)=(2^{p-1}-1)/p} is a Fermat quotient, with the consequence that p{textstyle p} divides the numerator of H(p−1)/2{displaystyle H_{(p-1)/2}} if and only if p{textstyle p} is a Wieferich prime.

In 1991, Eswarathasan and Levine^[7] defined J(p){displaystyle J(p)} as the set of all positive integers n{displaystyle n} such that the numerator of Hn{displaystyle H_{n}} is divisible by a prime number p.{displaystyle p.} They proved that

{p−1,p2−p,p2−1}⊆J(p){displaystyle {p-1,p^{2}-p,p^{2}-1}subseteq J(p)}

for all prime numbers p≥5,{displaystyle pgeq 5,} and they defined harmonic primes to be the primes p{textstyle p} such that J(p){displaystyle J(p)} has exactly 3 elements.

Eswarathasan and Levine also conjectured that J(p){displaystyle J(p)} is a finite set for all primes p,{displaystyle p,} and that there are infinitely many harmonic primes. Boyd^[8] verified that J(p){displaystyle J(p)} is finite for all prime numbers up to p=547{displaystyle p=547} except 83, 127, and 397; and he gave an heuristic suggesting that the density of the harmonic primes in the set of all primes should be 1/e.{displaystyle 1/e.} Sanna^[9] showed that Jp{displaystyle J_{p}} has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen^[10] proved that the number of elements of Jp{displaystyle J_{p}} not exceeding x{displaystyle x} is at most 3x23+125log⁡p{displaystyle 3x^{{frac {2}{3}}+{frac {1}{25log p}}}}, for all x≥1{displaystyle xgeq 1}.

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function

ψ(n)=Hn−1−γ.{displaystyle psi (n)=H_{n-1}-gamma .}

This relation is also frequently used to define the extension of the harmonic numbers to non-integer n. The harmonic numbers are also frequently used to define γ using the limit introduced earlier:

γ=limn→∞(Hn−ln⁡(n)),{displaystyle gamma =lim _{nrightarrow infty }{left(H_{n}-ln(n)right)},}

although

γ=limn→∞(Hn−ln⁡(n+12)){displaystyle gamma =lim _{nto infty }{left(H_{n}-ln left(n+{frac {1}{2}}right)right)}}

converges more quickly.

In 2002, Jeffrey Lagarias proved^[11] that the Riemann hypothesis is equivalent to the statement that

σ(n)≤Hn+(log⁡Hn)eHn,{displaystyle sigma (n)leq H_{n}+(log H_{n})e^{H_{n}},}

is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n.

The eigenvalues of the nonlocal problem

λφ(x)=∫−11φ(x)−φ(y)|x−y|dy{displaystyle lambda varphi (x)=int _{-1}^{1}{frac {varphi (x)-varphi (y)}{|x-y|}},dy}

are given by λ=2Hn{displaystyle lambda =2H_{n}}, where by convention, H0=0.{displaystyle H_{0}=0.}

Generalizations

Generalized harmonic numbers

The generalized harmonic number of order m of n is given by

Hn,m=∑k=1n1km.{displaystyle H_{n,m}=sum _{k=1}^{n}{frac {1}{k^{m}}}.}

The limit as n tends to infinity is finite if m > 1.

Other notations occasionally used include

Hn,m=Hn(m)=Hm(n).{displaystyle H_{n,m}=H_{n}^{(m)}=H_{m}(n).}

The special case of m = 0 gives Hn,0=n.{displaystyle H_{n,0}=n.} The special case of m = 1 is simply called a harmonic number and is frequently written without the m, as

Hn=∑k=1n1k.{displaystyle H_{n}=sum _{k=1}^{n}{frac {1}{k}}.}

The smallest natural number k such that kⁿ does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H′(k, n) is, for n=1, 2, ... :

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... (sequence A128670 in the OEIS)

In the limit as n → ∞ for m > 1, the generalized harmonic number converges to the Riemann zeta function

limn→∞Hn,m=ζ(m).{displaystyle lim _{nrightarrow infty }H_{n,m}=zeta (m).}

The related sum ∑k=1nkm{displaystyle sum _{k=1}^{n}k^{m}} occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

Some integrals of generalized harmonic numbers are

∫0aHx,2dx=aπ26−Ha{displaystyle int _{0}^{a}H_{x,2},dx=a{frac {pi ^{2}}{6}}-H_{a}}

and

∫0aHx,3dx=aA−12Ha,2,{displaystyle int _{0}^{a}H_{x,3},dx=aA-{frac {1}{2}}H_{a,2},} where A is Apéry's constant, i.e. ζ(3).

and

∑k=1nHk,m=(n+1)Hn,m−Hn,m−1 for m≥0{displaystyle sum _{k=1}^{n}H_{k,m}=(n+1)H_{n,m}-H_{n,m-1}{text{ for }}mgeq 0}

Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:

Hn,m=∑k=1n−1Hk,m−1k(k+1)+Hn,m−1n{displaystyle H_{n,m}=sum _{k=1}^{n-1}{frac {H_{k,m-1}}{k(k+1)}}+{frac {H_{n,m-1}}{n}}}   for example: H4,3=H1,21⋅2+H2,22⋅3+H3,23⋅4+H4,24{displaystyle H_{4,3}={frac {H_{1,2}}{1cdot 2}}+{frac {H_{2,2}}{2cdot 3}}+{frac {H_{3,2}}{3cdot 4}}+{frac {H_{4,2}}{4}}}

A generating function for the generalized harmonic numbers is

∑n=1∞znHn,m=Lim⁡(z)1−z,{displaystyle sum _{n=1}^{infty }z^{n}H_{n,m}={frac {operatorname {Li} _{m}(z)}{1-z}},}

where Lim⁡(z){displaystyle operatorname {Li} _{m}(z)} is the polylogarithm, and |z| < 1. The generating function given above for m = 1 is a special case of this formula.

A fractional argument for generalized harmonic numbers can be introduced as follows:

For every p,q>0{displaystyle p,q>0} integer, and m>1{displaystyle m>1} integer or not, we have from polygamma functions:

Hq/p,m=ζ(m)−pm∑k=1∞1(q+pk)m{displaystyle H_{q/p,m}=zeta (m)-p^{m}sum _{k=1}^{infty }{frac {1}{(q+pk)^{m}}}}

where ζ(m){displaystyle zeta (m)} is the Riemann zeta function. The relevant recurrence relation is:

Ha,m=Ha−1,m+1am{displaystyle H_{a,m}=H_{a-1,m}+{frac {1}{a^{m}}}}

Some special values are:

H14,2=16−8G−56π2{displaystyle H_{{frac {1}{4}},2}=16-8G-{tfrac {5}{6}}pi ^{2}} where G is Catalan's constant

H12,2=4−π23{displaystyle H_{{frac {1}{2}},2}=4-{tfrac {pi ^{2}}{3}}}

H34,2=8G+169−56π2{displaystyle H_{{frac {3}{4}},2}=8G+{tfrac {16}{9}}-{tfrac {5}{6}}pi ^{2}}

H14,3=64−27ζ(3)−π3{displaystyle H_{{frac {1}{4}},3}=64-27zeta (3)-pi ^{3}}

H12,3=8−6ζ(3){displaystyle H_{{frac {1}{2}},3}=8-6zeta (3)}

H34,3=(43)3−27ζ(3)+π3{displaystyle H_{{frac {3}{4}},3}={({tfrac {4}{3}})}^{3}-27zeta (3)+pi ^{3}}

Multiplication formulas

The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain

H2x=12(Hx+Hx−12)+ln⁡2{displaystyle H_{2x}={frac {1}{2}}left(H_{x}+H_{x-{frac {1}{2}}}right)+ln 2}

H3x=13(Hx+Hx−13+Hx−23)+ln⁡3,{displaystyle H_{3x}={frac {1}{3}}left(H_{x}+H_{x-{frac {1}{3}}}+H_{x-{frac {2}{3}}}right)+ln 3,}

or, more generally,

Hnx=1n(Hx+Hx−1n+Hx−2n+⋯+Hx−n−1n)+ln⁡n.{displaystyle H_{nx}={frac {1}{n}}left(H_{x}+H_{x-{frac {1}{n}}}+H_{x-{frac {2}{n}}}+cdots +H_{x-{frac {n-1}{n}}}right)+ln n.}

For generalized harmonic numbers, we have

H2x,2=12(ζ(2)+12(Hx,2+Hx−12,2)){displaystyle H_{2x,2}={frac {1}{2}}left(zeta (2)+{frac {1}{2}}left(H_{x,2}+H_{x-{frac {1}{2}},2}right)right)}

H3x,2=19(6ζ(2)+Hx,2+Hx−13,2+Hx−23,2),{displaystyle H_{3x,2}={frac {1}{9}}left(6zeta (2)+H_{x,2}+H_{x-{frac {1}{3}},2}+H_{x-{frac {2}{3}},2}right),}

where ζ(n){displaystyle zeta (n)} is the Riemann zeta function.

Hyperharmonic numbers

The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers.^[1]:258 Let

Hn(0)=1n.{displaystyle H_{n}^{(0)}={frac {1}{n}}.}

Then the nth hyperharmonic number of order r (r>0) is defined recursively as

Hn(r)=∑k=1nHk(r−1).{displaystyle H_{n}^{(r)}=sum _{k=1}^{n}H_{k}^{(r-1)}.}

In particular, Hn(1){displaystyle H_{n}^{(1)}} is the ordinary harmonic number Hn{displaystyle H_{n}}.

Harmonic numbers for real and complex values

The formulae given above,

Hx=∫011−tx1−tdt=−∑k=1∞(xk)(−1)kk{displaystyle H_{x}=int _{0}^{1}{frac {1-t^{x}}{1-t}},dt=-sum _{k=1}^{infty }{x choose k}{frac {(-1)^{k}}{k}}}

are an integral and a series representation for a function that interpolates the harmonic numbers and, via analytic continuation, extends the definition to the complex plane other than the negative integers x. The interpolating function is in fact closely related to the digamma function

Hx=ψ(x+1)+γ,{displaystyle H_{x}=psi (x+1)+gamma ,}

where ψ(x) is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

Hx,2=−∑k=1∞(−1)kk(xk)Hk.{displaystyle H_{x,2}=-sum _{k=1}^{infty }{frac {(-1)^{k}}{k}}{x choose k}H_{k}.}

The Taylor series for the harmonic numbers is

Hx=∑k=2∞(−1)kζ(k)xk−1 for |x|<1{displaystyle H_{x}=sum _{k=2}^{infty }(-1)^{k}zeta (k);x^{k-1}quad {text{ for }}|x|<1}

which comes from the Taylor series for the digamma function.

Alternative, asymptotic formulation

When seeking to approximate H_x for a complex number x it turns out that it is effective to first compute H_m for some large integer m, then use that to approximate a value for H_m+x, and then use the recursion relation H_n = H_n−1 + 1/n backwards m times, to unwind it to an approximation for H_x. Furthermore, this approximation is exact in the limit as m goes to infinity.

Specifically, for every integer n, we have that

limm→∞[Hm−Hm+n]=0,{displaystyle lim _{mrightarrow infty }left[H_{m}-H_{m+n}right]=0,,}

and we can ask that the formula be obeyed if the arbitrary integer n is replaced by an arbitrary complex number x

limm→∞[Hm−Hm+x]=0.{displaystyle lim _{mrightarrow infty }left[H_{m}-H_{m+x}right]=0,.}

Adding H_x to both sides gives

Hx=limm→∞[Hm−(Hm+x−Hx)]=limm→∞[(∑k=1m1k)−(∑k=1m1x+k)]=limm→∞∑k=1m(1k−1x+k)=x∑k=1∞1k(x+k).{displaystyle {begin{aligned}H_{x}&=lim _{mrightarrow infty }left[H_{m}-(H_{m+x}-H_{x})right]\[6pt]&=lim _{mrightarrow infty }left[left(sum _{k=1}^{m}{frac {1}{k}}right)-left(sum _{k=1}^{m}{frac {1}{x+k}}right)right]\[6pt]&=lim _{mrightarrow infty }sum _{k=1}^{m}left({frac {1}{k}}-{frac {1}{x+k}}right)=xsum _{k=1}^{infty }{frac {1}{k(x+k)}},.end{aligned}}}

This last expression for H_x is well defined for any complex number x except the negative integers, which fail because trying to use the recursion relation H_n = H_n−1 + 1/n backwards through the value n = 0 involves a division by zero. By construction, the function H_x is the unique function of x for which (1) H₀ = 0, (2) H_x = H_x−1 + 1/x for all complex values x except the non-positive integers, and (3) lim_m→+∞ (H_m+x − H_m) = 0 for all complex values x.

Based on this last formula, it can be shown that:

∫01Hxdx=γ,{displaystyle int _{0}^{1}H_{x},dx=gamma ,,}

where γ is the Euler–Mascheroni constant or, more generally, for every n we have:

∫0nHxdx=nγ+ln⁡(n!).{displaystyle int _{0}^{n}H_{x},dx=ngamma +ln {(n!)},.}

Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral

Hα=∫011−xα1−xdx.{displaystyle H_{alpha }=int _{0}^{1}{frac {1-x^{alpha }}{1-x}},dx,.}

More values may be generated from the recurrence relation

Hα=Hα−1+1α,{displaystyle H_{alpha }=H_{alpha -1}+{frac {1}{alpha }},,}

or from the reflection relation

H1−α−Hα=πcot⁡(πα)−1α+11−α.{displaystyle H_{1-alpha }-H_{alpha }=pi cot {(pi alpha )}-{frac {1}{alpha }}+{frac {1}{1-alpha }},.}

For example:

H12=2−2ln⁡2{displaystyle H_{frac {1}{2}}=2-2ln {2}}

H13=3−π23−32ln⁡3{displaystyle H_{frac {1}{3}}=3-{tfrac {pi }{2{sqrt {3}}}}-{tfrac {3}{2}}ln {3}}

H23=32(1−ln⁡3)+3π6{displaystyle H_{frac {2}{3}}={tfrac {3}{2}}(1-ln {3})+{sqrt {3}}{tfrac {pi }{6}}}

H14=4−π2−3ln⁡2{displaystyle H_{frac {1}{4}}=4-{tfrac {pi }{2}}-3ln {2}}

H34=43−3ln⁡2+π2{displaystyle H_{frac {3}{4}}={tfrac {4}{3}}-3ln {2}+{tfrac {pi }{2}}}

H16=6−π23−2ln⁡2−32ln⁡3{displaystyle H_{frac {1}{6}}=6-{tfrac {pi }{2}}{sqrt {3}}-2ln {2}-{tfrac {3}{2}}ln {3}}

H18=8−π2−4ln⁡2−12{π+ln⁡(2+2)−ln⁡(2−2)}{displaystyle H_{frac {1}{8}}=8-{tfrac {pi }{2}}-4ln {2}-{tfrac {1}{sqrt {2}}}left{pi +ln left(2+{sqrt {2}}right)-ln left(2-{sqrt {2}}right)right}}

H112=12−3(ln⁡2+ln⁡32)−π(1+32)+23ln⁡(2−3){displaystyle H_{frac {1}{12}}=12-3left(ln {2}+{tfrac {ln {3}}{2}}right)-pi left(1+{tfrac {sqrt {3}}{2}}right)+2{sqrt {3}}ln left({sqrt {2-{sqrt {3}}}}right)}

For positive integers p and q with p < q, we have:

Hpq=qp+2∑k=1⌊q−12⌋cos⁡(2πpkq)ln⁡(sin⁡(πkq))−π2cot⁡(πpq)−ln⁡(2q){displaystyle H_{frac {p}{q}}={frac {q}{p}}+2sum _{k=1}^{lfloor {frac {q-1}{2}}rfloor }cos left({frac {2pi pk}{q}}right)ln left({sin left({frac {pi k}{q}}right)}right)-{frac {pi }{2}}cot left({frac {pi p}{q}}right)-ln left(2qright)}

Relation to the Riemann zeta function

Some derivatives of fractional harmonic numbers are given by:

dnHxdxn=(−1)n+1n![ζ(n+1)−Hx,n+1]dnHx,2dxn=(−1)n+1(n+1)![ζ(n+2)−Hx,n+2]dnHx,3dxn=(−1)n+112(n+2)![ζ(n+3)−Hx,n+3].{displaystyle {begin{aligned}{frac {d^{n}H_{x}}{dx^{n}}}&=(-1)^{n+1}n!left[zeta (n+1)-H_{x,n+1}right]\[6pt]{frac {d^{n}H_{x,2}}{dx^{n}}}&=(-1)^{n+1}(n+1)!left[zeta (n+2)-H_{x,n+2}right]\[6pt]{frac {d^{n}H_{x,3}}{dx^{n}}}&=(-1)^{n+1}{frac {1}{2}}(n+2)!left[zeta (n+3)-H_{x,n+3}right].end{aligned}}}

And using Maclaurin series, we have for x < 1:

Hx=∑n=1∞(−1)n+1xnζ(n+1)Hx,2=∑n=1∞(−1)n+1(n+1)xnζ(n+2)Hx,3=12∑n=1∞(−1)n+1(n+1)(n+2)xnζ(n+3).{displaystyle {begin{aligned}H_{x}&=sum _{n=1}^{infty }(-1)^{n+1}x^{n}zeta (n+1)\[5pt]H_{x,2}&=sum _{n=1}^{infty }(-1)^{n+1}(n+1)x^{n}zeta (n+2)\[5pt]H_{x,3}&={frac {1}{2}}sum _{n=1}^{infty }(-1)^{n+1}(n+1)(n+2)x^{n}zeta (n+3).end{aligned}}}

For fractional arguments between 0 and 1, and for a > 1:

H1/a=1a(ζ(2)−1aζ(3)+1a2ζ(4)−1a3ζ(5)+⋯)H1/a,2=1a(2ζ(3)−3aζ(4)+4a2ζ(5)−5a3ζ(6)+⋯)H1/a,3=12a(2⋅3ζ(4)−3⋅4aζ(5)+4⋅5a2ζ(6)−5⋅6a3ζ(7)+⋯).{displaystyle {begin{aligned}H_{1/a}&={frac {1}{a}}left(zeta (2)-{frac {1}{a}}zeta (3)+{frac {1}{a^{2}}}zeta (4)-{frac {1}{a^{3}}}zeta (5)+cdots right)\[6pt]H_{1/a,,2}&={frac {1}{a}}left(2zeta (3)-{frac {3}{a}}zeta (4)+{frac {4}{a^{2}}}zeta (5)-{frac {5}{a^{3}}}zeta (6)+cdots right)\[6pt]H_{1/a,,3}&={frac {1}{2a}}left(2cdot 3zeta (4)-{frac {3cdot 4}{a}}zeta (5)+{frac {4cdot 5}{a^{2}}}zeta (6)-{frac {5cdot 6}{a^{3}}}zeta (7)+cdots right).end{aligned}}}

See also

• Watterson estimator


• Tajima's D


• Coupon collector's problem


• Jeep problem


• Riemann zeta function


• List of sums of reciprocals

Notes

References

• 
  Arthur T. Benjamin; Gregory O. Preston; Jennifer J. Quinn (2002). "A Stirling Encounter with Harmonic Numbers" (PDF). Mathematics Magazine. 75 (2): 95–103. doi:10.2307/3219141.
  


• 
  Donald Knuth (1997). "Section 1.2.7: Harmonic Numbers". The Art of Computer Programming. Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley. pp. 75–79. ISBN 0-201-89683-4.
  


• Ed Sandifer, How Euler Did It — Estimating the Basel problem (2003)


• 
  Paule, Peter; Schneider, Carsten (2003). "Computer Proofs of a New Family of Harmonic Number Identities" (PDF). Adv. Appl. Math. 31 (2): 359–378. doi:10.1016/s0196-8858(03)00016-2.
  


• 
  Wenchang Chu (2004). "A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers" (PDF). The Electronic Journal of Combinatorics. 11: N15.
  


• 
  Ayhan Dil; István Mező (2008). "A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers". Applied Mathematics and Computation. 206 (2): 942–951. arXiv:0803.4388. doi:10.1016/j.amc.2008.10.013.
  

External links

• Weisstein, Eric W. "Harmonic Number". MathWorld.

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